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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 我们可以从概率分布中抽样。\n",
    "* 我们可以使用联合分布、条件分布、贝叶斯定理、边际化和独立性假设来分析多个随机变量。\n",
    "* 期望和方差为概括概率分布的关键特征提供了使用的度量形式。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2.6.1 概率论的基本概论"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "import os \n",
    "os.environ[\"KMP_DUPLICATE_LIB_OK\"]=\"TRUE\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import torch\n",
    "from torch.distributions import multinomial\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0., 1., 0., 0., 0., 0.])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fair_probs = torch.ones([6]) / 6\n",
    "multinomial.Multinomial(1, fair_probs).sample()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([1., 2., 0., 1., 2., 4.])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "multinomial.Multinomial(10,fair_probs).sample()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([0.1600, 0.1730, 0.1650, 0.1670, 0.1630, 0.1720])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 将结果存储为32为浮点数以进行除法\n",
    "counts = multinomial.Multinomial(1000,fair_probs).sample()\n",
    "counts / 1000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "```mermaid\n",
    "flowchart TD\n",
    "    A[开始] --> B[生成 500 组样本数据]\n",
    "    B --> C[计算累计计数]\n",
    "    C --> D[计算概率估计]\n",
    "    D --> E[设置图形大小]\n",
    "    E --> F[绘制每个面的概率估计曲线]\n",
    "    F --> G[添加参考线和标签]\n",
    "    G --> H[显示图形]\n",
    "    H --> I[结束]\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
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     },
     "metadata": {},
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    }
   ],
   "source": [
    "counts = multinomial.Multinomial(10,fair_probs).sample((500,))\n",
    "cum_counts = counts.cumsum(dim=0)\n",
    "estimates = cum_counts / cum_counts.sum(dim=1,keepdim=True)\n",
    "\n",
    "d2l.set_figsize((6,4))\n",
    "for i in range(6):\n",
    "    d2l.plt.plot(estimates[:,i].numpy(), label=(\"P(die=\" + str(i+1) + \")\"))\n",
    "d2l.plt.axhline(y = 0.167, color='black', linestyle='dashed')\n",
    "d2l.plt.gca().set_xlabel('Groups of experiments')\n",
    "d2l.plt.gca().set_ylabel('Estimated probability')\n",
    "d2l.plt.legend()\n",
    "d2l.plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 概率论公理"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "概率具有以下属性：\n",
    "* 对于任意事件$A$，其概率不会是负数，即$P(A)\\ge 0$。\n",
    "* 整个样本空间的概率为$1$，即$P(S)=1$。\n",
    "* 对于互斥事件的任意一个可数序列$A_1,A_2,\\cdots,A_n$，有$P(\\bigcup_{i=1}^n A_i)=\\sum_{i=1}^n P(A_i)$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 随机变量"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "离散随机变量和连续随机变量之间存在微妙的区别。我们将这个看到某个数值的可能性量化为密度"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2.6.2 处理多个随机变量"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 联合概率"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "给定任意值$a$和$b$，联合概率可以回答：$A = a $和$B = b$同时发生的概率是多少。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 条件概率"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$0 \\le \\frac{P(A = a,B = b)}{P(A = a)} \\le 1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 贝叶斯定理"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$P(A|B) = \\frac{P(B|A)P(A)}{P(B)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 边际化\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$P(B) = \\sum _A P(A,B)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 独立性"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如果两个随机变量$A$和$B$是独立的，意味着事件$A$的发生跟事件$B$的发生无关，符号为$A\\perp B$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$P(A|B) = \\frac {P(A,B)} {P(B)} = P(A)$等价于$P(A,B)=P(A)P(B)$,因此两个随机变量是独立的，当且仅当两个随机变量的联合分布是其各自分布的乘积，同样地，给定另一个随机变量$C$时，两个随机变量$A$和$B$时条件独立的，当且仅当$P(A,B|C)=P(A|C)P(B|C)$.符号为$A\\perp B|C$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2.6.3 期望和方差"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "期望："
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$E[X] = \\sum_{i=1} xP(X=x)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(x)$的输入是从分布$P$中抽取的随机变量时，$f(x)$的期望："
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$E_{x~P}[f(x)] = \\sum _x p(x)f(x)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "衡量随机变量$X$与其期望的偏差："
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$Var[X] = E[X - E[X] ^ 2] = E[X^2] - E[X]^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "函数值偏离该函数的期望的程度："
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$Var[f(x)] = E[(f(x) - E[f(x)])^2]$$"
   ]
  }
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